I may be late to the party here, and I can't say for sure as I am only a senior with little computational chem knowledge, however this is the kind of stuff I think and read about in my free time. I am not 100% sure what a Gaussian basis is as I am in Intro to Linear Algebra not, so it's a little beyond the scope of that course, but I would assume its about a basis for a distribution space similar to a Gaussian distribution (you know, what the curving of your undergraduate grades are based off of). Well anyway from quantum you should have learned or will learn that the position and momentum of an electron can only be known with so much certainty, the greater we know what one of them is, the less we know about another. So for DFT they are calculating the probability of the electron being in a certain place, as they map all of this, you get a map of where the electron can be with some places more likely, i.e. more electron probability density. So there is a maximum, with decreasing density on either side of the maximum of the electron being found there. So from what I understand it is in a probability space not a Rn space.
Just my guess from what knowledge I have and my intuition.